Theorem r0cld 23126

Description: The analogue of the T₁ axiom (singletons are closed) for an R₀ space. In an R₀ space the set of all points topologically indistinguishable from 𝐴 is closed. (Contributed by Mario Carneiro, 25-Aug-2015.)

Hypothesis

Ref	Expression
kqval.2	⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})

Assertion

Ref	Expression
r0cld	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → {𝑧 ∈ 𝑋 ∣ ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝐴 ∈ 𝑜)} ∈ (Clsd‘𝐽))

Distinct variable groups:   𝑥,𝑜,𝑦,𝑧,𝐴   𝑜,𝐽,𝑥,𝑦,𝑧   𝑜,𝐹,𝑧   𝑜,𝑋,𝑥,𝑦,𝑧

Allowed substitution hints:   𝐹(𝑥,𝑦)

Proof of Theorem r0cld

Step	Hyp	Ref	Expression
1		kqval.2	. . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})
2	1	kqffn 23113	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
3	2	3ad2ant1 1133	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → 𝐹 Fn 𝑋)
4		fncnvima2 7016	. . . 4 ⊢ (𝐹 Fn 𝑋 → (◡𝐹 “ {(𝐹‘𝐴)}) = {𝑧 ∈ 𝑋 ∣ (𝐹‘𝑧) ∈ {(𝐹‘𝐴)}})
5	3, 4	syl 17	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → (◡𝐹 “ {(𝐹‘𝐴)}) = {𝑧 ∈ 𝑋 ∣ (𝐹‘𝑧) ∈ {(𝐹‘𝐴)}})
6		fvex 6860	. . . . . 6 ⊢ (𝐹‘𝑧) ∈ V
7	6	elsn 4606	. . . . 5 ⊢ ((𝐹‘𝑧) ∈ {(𝐹‘𝐴)} ↔ (𝐹‘𝑧) = (𝐹‘𝐴))
8		simpl1 1191	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
9		simpr 485	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ 𝑋)
10		simpl3 1193	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) → 𝐴 ∈ 𝑋)
11	1	kqfeq 23112	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝐹‘𝑧) = (𝐹‘𝐴) ↔ ∀𝑦 ∈ 𝐽 (𝑧 ∈ 𝑦 ↔ 𝐴 ∈ 𝑦)))
12		eleq2w 2816	. . . . . . . . 9 ⊢ (𝑦 = 𝑜 → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑜))
13		eleq2w 2816	. . . . . . . . 9 ⊢ (𝑦 = 𝑜 → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ 𝑜))
14	12, 13	bibi12d 345	. . . . . . . 8 ⊢ (𝑦 = 𝑜 → ((𝑧 ∈ 𝑦 ↔ 𝐴 ∈ 𝑦) ↔ (𝑧 ∈ 𝑜 ↔ 𝐴 ∈ 𝑜)))
15	14	cbvralvw 3223	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝐽 (𝑧 ∈ 𝑦 ↔ 𝐴 ∈ 𝑦) ↔ ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝐴 ∈ 𝑜))
16	11, 15	bitrdi 286	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝐹‘𝑧) = (𝐹‘𝐴) ↔ ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝐴 ∈ 𝑜)))
17	8, 9, 10, 16	syl3anc 1371	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) → ((𝐹‘𝑧) = (𝐹‘𝐴) ↔ ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝐴 ∈ 𝑜)))
18	7, 17	bitrid 282	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) → ((𝐹‘𝑧) ∈ {(𝐹‘𝐴)} ↔ ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝐴 ∈ 𝑜)))
19	18	rabbidva 3412	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → {𝑧 ∈ 𝑋 ∣ (𝐹‘𝑧) ∈ {(𝐹‘𝐴)}} = {𝑧 ∈ 𝑋 ∣ ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝐴 ∈ 𝑜)})
20	5, 19	eqtrd 2771	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → (◡𝐹 “ {(𝐹‘𝐴)}) = {𝑧 ∈ 𝑋 ∣ ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝐴 ∈ 𝑜)})
21	1	kqid 23116	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
22	21	3ad2ant1 1133	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
23		simp2 1137	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → (KQ‘𝐽) ∈ Fre)
24		simp3 1138	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋)
25		fnfvelrn 7036	. . . . . 6 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐹‘𝐴) ∈ ran 𝐹)
26	3, 24, 25	syl2anc 584	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → (𝐹‘𝐴) ∈ ran 𝐹)
27	1	kqtopon 23115	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
28	27	3ad2ant1 1133	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
29		toponuni 22300	. . . . . 6 ⊢ ((KQ‘𝐽) ∈ (TopOn‘ran 𝐹) → ran 𝐹 = ∪ (KQ‘𝐽))
30	28, 29	syl 17	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → ran 𝐹 = ∪ (KQ‘𝐽))
31	26, 30	eleqtrd 2834	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → (𝐹‘𝐴) ∈ ∪ (KQ‘𝐽))
32		eqid 2731	. . . . 5 ⊢ ∪ (KQ‘𝐽) = ∪ (KQ‘𝐽)
33	32	t1sncld 22714	. . . 4 ⊢ (((KQ‘𝐽) ∈ Fre ∧ (𝐹‘𝐴) ∈ ∪ (KQ‘𝐽)) → {(𝐹‘𝐴)} ∈ (Clsd‘(KQ‘𝐽)))
34	23, 31, 33	syl2anc 584	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → {(𝐹‘𝐴)} ∈ (Clsd‘(KQ‘𝐽)))
35		cnclima 22656	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn (KQ‘𝐽)) ∧ {(𝐹‘𝐴)} ∈ (Clsd‘(KQ‘𝐽))) → (◡𝐹 “ {(𝐹‘𝐴)}) ∈ (Clsd‘𝐽))
36	22, 34, 35	syl2anc 584	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → (◡𝐹 “ {(𝐹‘𝐴)}) ∈ (Clsd‘𝐽))
37	20, 36	eqeltrrd 2833	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → {𝑧 ∈ 𝑋 ∣ ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝐴 ∈ 𝑜)} ∈ (Clsd‘𝐽))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3060  {crab 3405  {csn 4591  ∪ cuni 4870   ↦ cmpt 5193  ^◡ccnv 5637  ran crn 5639   “ cima 5641   Fn wfn 6496  ‘cfv 6501  (class class class)co 7362  TopOnctopon 22296  Clsdccld 22404   Cn ccn 22612  Frect1 22695  KQckq 23081

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-map 8774  df-qtop 17403  df-top 22280  df-topon 22297  df-cld 22407  df-cn 22615  df-t1 22702  df-kq 23082

This theorem is referenced by:  nrmr0reg  23137